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Unformatted text preview: ibuted amount of time, ﬁnd the stationary distribution of the number of
children playing or in line at each of the two machines. 158 CHAPTER 4. CONTINUOUS TIME MARKOV CHAINS Chapter 5 Martingales
In this chapter we will introduce a class of process that can be thought of as the
fortune of a gambler betting on a fair game. These results will be important
when we consider applications to ﬁnance in the next chapter. In addition, they
will allow us to give more transparent proofs of some facts from Chapter 1
concerning exit distributions and exit times for Markov chains. 5.1 Conditional Expectation Our study of martingales will rely heavily on the notion of conditional expectation and involve some formulas that may not be familiar, so we will review
them here. We begin with several deﬁnitions. Given an event A we deﬁne its
0 x 2 Ac
In words, 1A is “1 on A” (and 0 otherwise). Given a random variable Y , we
deﬁne the integral of Y over A to be
E (Y ; A) = E (Y 1A )
Note that multiply...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).
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